Suppose $x$ and $y$ are integers and $x^2$ is a multiple of $y^2$. Is $x$ necessarily a multiple of $y$?

Question

Suppose $x$ and $y$ are integers and $x^2$ is a multiple of $y^2$. Is $x$ necessarily a multiple of $y$?

Came across this question as part of a homework set. I tried the following: Suppose $x^2 \,\vert\, y^2$, then $x^2$ $=$ $ky^2$ for some k. Taking square roots on both sides, we end up with $x$ $=$ $\pm$$(\sqrt{k})$$y$.

However, I am unsure of how to proceed from this point. I believe we either need to prove or disprove the fact that $\sqrt{k}$ is an integer value for all values of $k$ but the method eludes me.

If $x^2|y^2$ then $y^2=kx^2$, not the other way around. It doesn't change the argument. — Ross Millikan, Mar 02 '20 at 16:04

score 1 · Answer 1 · answered Mar 02 '20 at 15:58

Hint

Divide the equation $x^2=ky^2$ by the square of the gcd of $x,y$. You get $X^2=kY^2$ where $X,Y$ are coprime.

Then prove that this equation is only possible if $k$ is a square.

You can finally conclude that $x^2$ is a multiple of $y^2$ if and only if $x$ is a multiple of $y$.

score 0 · Answer 2 · answered Mar 02 '20 at 15:55

0

If there exists some integer $d$ such that $x^2=dy^2$, I claim there is some integer $k$ such that $k^2=d$. The claim is proved here.

So, $x=ky$.

answered Mar 02 '20 at 15:55

Rushabh Mehta

13,845

score 0 · Answer 3 · answered Mar 02 '20 at 16:30

0

Hint:

If $k$ is not a square, is $\sqrt ky$ an integer?

answered Mar 02 '20 at 16:30

cansomeonehelpmeout

14,020
3
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Suppose $x$ and $y$ are integers and $x^2$ is a multiple of $y^2$. Is $x$ necessarily a multiple of $y$?

3 Answers3